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United States Patent |
5,285,159
|
Bodenhausen
,   et al.
|
February 8, 1994
|
RF pulse cascade for the generation of NMR spectra
Abstract
A method for generating a spectrum of NMR signals with rectangular
characteristic in the frequency space by radiating a sequence of RF pulses
onto a sample in the homogeneous magnetic field is presented. The RF
pulses of the sequence are amplitude modulated and the enveloping function
of the sequence consists of a superposition of bell-curve-shaped RF pulses
with optimized positions of the maxima, pulse widths and peak amplitudes.
The response signals, in dependence on the standardized frequency, shift,
approximate virtually ideally, a rectangular function. The method is
applicable with success especially in image generation for NMR tomography,
in multidimensional NMR spectroscopy, as well as in volume selective NMR
spectroscopy.
Inventors:
|
Bodenhausen; Geoffrey (Pully, CH);
Emsley; Lyndon (Lausanne, CH)
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Assignee:
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Spectrospin AG, Ind. (Fallanden);
Bruker Analytische Messtechnik GmbH (Rheinstetten)
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Appl. No.:
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852235 |
Filed:
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June 1, 1992 |
PCT Filed:
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October 6, 1990
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PCT NO:
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PCT/EP90/01681
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371 Date:
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June 1, 1992
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102(e) Date:
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June 1, 1992
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PCT PUB.NO.:
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WO91/09322 |
PCT PUB. Date:
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June 27, 1991 |
Foreign Application Priority Data
Current U.S. Class: |
324/314; 324/307 |
Intern'l Class: |
G01V 003/00 |
Field of Search: |
324/314,300,307,309,311,312
|
References Cited
U.S. Patent Documents
4682106 | Jul., 1987 | Vatis et al. | 324/307.
|
4710718 | Dec., 1987 | Shaka | 324/314.
|
4733186 | Mar., 1988 | Oppelt | 324/309.
|
4746863 | May., 1988 | Crooks et al. | 324/309.
|
4757260 | Jul., 1988 | Tsuda | 324/309.
|
4774467 | Sep., 1988 | Sorensen | 324/311.
|
4818940 | Apr., 1989 | Hennig | 324/309.
|
4878021 | Oct., 1989 | Granot | 324/309.
|
5126671 | Jun., 1992 | Bodenhausen et al. | 324/314.
|
Other References
Proceedings of the Fourth Annual Meeting of the Society of Magnetic
Resonance in Medicine, London, Aug. 1985, p. 958 ff, Conolly et al.:
Selective Pulse Design Via Optimal Control Theory.
Journal of Magnetic Resonance, 74, 226-263 (1987) Murdoch et al.:
Computer-Optimized Narrowband Pulses for Multislice Imaging.
Magnetic Resonance in Medicine, 5, 217-237 (1987) Nago et al.: General
Solution to the NMR Excitation Problem for Noninteracting Spins.
|
Primary Examiner: Arana; Louis
Attorney, Agent or Firm: Vigil; Thomas R.
Claims
We claim:
1. A method for generating a spectrum of nuclear magnetic resonance signals
comprising the steps of:
a) selecting a pulse sequence with a number n of at least 2 but not more
than 10 amplitude modulated bell shaped RF pulses, each RF pulse
comprising an amplitude w.sub.k.sup.max, an extremum position
t.sub.k.sup.max, and a width a.sub.k ;
b) varying the amplitude, extremum position, and width of each pulse to
minimize a deviation of a signal response frequency space envelop from a
rectangular function:
c) radiating the pulse sequence onto a sample located in a homogeneous
static magnetic field to produce a nuclear magnetic resonance signal
response with an approximately rectangular frequency space envelope.
2. The method according to claim 1, wherein the minimization of the
deviation of the envelope is accomplished by use of a numerical fit
program on an automatic data processing installation.
3. The method according to claim 1, wherein the minimization of the
deviation of the envelope is accomplished by optimization of a
predetermined error function, which describes the deviation of the
envelope from a rectangular function.
4. The method according to claim 3, wherein the minimization is stopped
when the absolute value of the error function exceeds a predetermined
limit value.
5. The method according to claim 1, wherein the sequence contains RF pulses
whose amplitude distribution follows the form of a Lorentz curve.
6. The method according to claim 1, wherein the sequence contains RF pulses
whose amplitude distribution follows the form of a Gaussian curve.
7. The method according to claim 1, wherein n=3 and the method is used for
the selective inversion of nuclear magnetic moments.
8. The method according to claim 7, wherein the pulse sequence has a time
duration t.sub.p, a kth pulse has a half pulse width
.delta.t.sub.k.sup.1/2 at a half pulse amplitude
w.sub.k.sup.max /2, a.sub.k =ln2/(.delta.t.sub.k.sup.1/2).sup.2, and
w.sub.k.sup.max, t.sub.k.sup.max /t.sub.p and
.delta.t.sub.k.sup.1/2 /t.sub.p have values in ranges:
1.2<w.sub.2.sup.max /w.sub.1.sup.max <1.5; 0.3<w.sub.3.sup.max
/w.sub.1.sup.max <0.7
15<.delta.t.sub.1.sup.1/2 /t.sub.p <25; 15<.delta.t.sub.2.sup.1/2 /t.sub.p
<25; 20<.delta.t.sub.3.sup.1/2 /t.sub.p <30;
20<t.sub.1.sup.max /t.sub.p <40; 40<t.sub.2.sup.max /t.sub.p <60;
70<t.sub.3.sup.max /t.sub.p <90.
9. The method according to claim 8, wherein w.sub.k.sup.max,
t.sub.k.sup.max, and .delta.t.sub.k.sup.1/2 /t.sub.p have values:
w.sub.1.sup.max =-1.00; w.sub.2.sup.max =1.37; w.sub.3.sup.max =0.49;
.delta.t.sub.1.sup.1/2 /t.sub.p =18.9; .delta.t.sub.2.sup.1/2 /t.sub.p
=18.3; .delta.t.sub.3.sup.1/2 /t.sub.p =24.3;
t.sub.1.sup.max /t.sub.p =28.7; t.sub.2.sup.max /t.sub.p =50.8;
t.sub.3.sup.max /t.sub.p =79.5.
10. The method according to claim 1, wherein n=4, and the method is used
for an inphase excitation of transverse magnetization.
11. The method according to claim 10, wherein the pulse sequence has a time
duration t.sub.p, the kth pulse has a half pulse width
.delta.t.sub.k.sup.1/2 at a half pulse amplitude w.sub.k.sup.max /2,
a.sub.k =ln2/(.delta.t.sub.k.sup.1/2).sup.2, and w.sub.k.sup.max,
t.sub.k.sup.max /t.sub.p and .delta.t.sub.k.sup.1/2 /t.sub.p have values
in ranges:
0.95<w.sub.2.sup.max /w.sub.1.sup.max <1.35; -1.7<w.sub.3.sup.max
/w.sub.1.sup.max <-1.3; -0.7<w.sub.4.sup.max /w.sub.1.sup.max <-1.4;
15<.delta.t.sub.1.sup.1/2 /t.sub.p <20; 10<.delta.t.sub.2.sup.1/2 /t.sub.p
<15; 10<.delta.t.sub.3.sup.1/2 /t.sub.p 15; 10<.delta.t.sub.4.sup.1/2
/t.sub.p <20;
5<t.sub.1.sup.max /t.sub.p <25; 40<t.sub.2.sup.max /t.sub.p <60;
55<t.sub.3.sup.max /t.sub.p <75; 75<t.sub.4.sup.max /t.sub.p <95.
12. The method according to claim 11, wherein w.sub.k.sup.max,
t.sub.k.sup.max and .delta.t.sub.k.sup.1/2 /t.sub.p have values:
w.sub.1.sup.max =0.62; w.sub.2.sup.max =0.72; w.sub.3.sup.max =-0.91;
w.sub.4.sup.max =-0.33
.delta.t.sub.1.sup.1/2 /t.sub.p =17.2; .delta.t.sub.2.sup.1/2 /t.sub.p
=12.9; .delta.t.sub.3.sup.1/2 /t.sub.p =11.9;
.delta.t.sub.4.sup.1/2 /t.sub.p =13.9;
t.sub.1.sup.max /t.sub.p =17.7; t.sub.2.sup.max /t.sub.p =49.2;
t.sub.3.sup.max /t.sub.p =65.3;
t.sub.4.sup.max /t.sub.p =89.2.
13. The method according to claim 1 further comprising the step of:
d) generating section images for an NMR tomograph.
14. The method according to claim 1 further comprising the step of:
d) generating a multidimensional NMR spectroscopy.
15. The method according to claim 14, wherein the NMR spectroscopy is
volume selective.
16. The method according to claim 14, wherein the sequence of n RF pulses
is part of a NOESY pulse sequence.
17. The method according to claim 14, wherein the sequence of n RF pulses
is part of a COSY pulse sequence.
18. A nuclear magnetic resonance spectrometer with an RF pulse generator
and an arrangement for the drive of the RF pulse generator, the
arrangement comprising a memory for the storage of data sets for the
generation of RF pulse sequences wherein the memory comprises a data set
for the generation of a pulse sequence with a number n of at least two but
not more than 10 amplitude modulated bell shaped RF pulses, each RF pulse
comprising an amplitude w.sub.k.sup.max, an extremum position
t.sub.k.sup.max, and a width a.sub.k and, in advance, the amplitude
distribution of the entire pulse sequence has been determined in a
suitable optimization procedure under a criterion of a minimization of a
deviation of an envelope of a nuclear magnetic resonance signal response
in frequency space from a rectangular function by variation, for each
pulse, of w.sub.k.sup.max, t.sub.k.sup.max, and a.sub.k.
Description
The invention relates to a method for generating a spectrum of nuclear
magnetic resonance signals by radiating a sequence of n RF pulses onto a
sample which is present in a homogeneous static magnetic field, in which
the envelope of the nuclear magnetic resonance signal response in the
frequency space is approximately a rectangular function.
The capacity for the excitation or inversion of nuclear spins over a
selected band of frequencies has become an important part in many
experiments in modern nuclear magnetic resonance. In high resolving NMR
spectroscopy it is often desirable to restrict the width of the relevant
frequency domains, especially in two and three dimensional spectroscopy.
Frequency selective excitation is also an integral component of a whole
class of experiments in which two or three dimensional spectra are
compressed into one dimension. The perhaps most important application of
selective excitation are the methods for NMR imaging. In this context, two
especially important problems are to be mentioned, namely the inversion
over a well defined rectangular window in the frequency spectrum and the
excitation of transverse magnetization with minimal phase dispersion
proceeding from longitudinal magnetization, again over a rectangular
window in the frequency spectrum.
Among the numerous methods that were proposed by different authors for
rectangular inversion and excitation, a large number are based on the use
of composite pulses that consist of trains of hard pulses which are phase
shifted against one another, but have rectangular envelopes and constant
amplitudes. Other processes for rectangular inversion and excitation use
amplitude and/or phase modulation. In the further discussion there are to
be considered only pulses with purely amplitude modulated envelopes. For
the spin inversion, it is possible to use radio frequency (RF) pulses
whose envelopes can be described by simple analytic functions, such as,
for example, Gaussian, Sinc or Hermite functions. The only known amplitude
modulated pulse for in-phase excitation with an envelope that can be
described by analytical function is a Gaussian pulse with 270.degree.
on-resonance flip angle. Although with this pulse the above problems can
be partly solved, in the case of demanding applications, due to residual
phase dispersion and the deficient selectivity, there still remain
unsolved problems.
Latest strivings for the optimization of formed pulses were based on the
division of an arbitrary envelope in the time domain into many discrete
intervals and the variation of the RF amplitude in each individual
interval until the signal response approximates a target function. For
this purpose, various degrees of refinements were introduced, among
others, the optimal regulating technique of Conolly et al. in "Proceedings
of the Fourth Annual Meeting of the Society of Magnetic Resonance in
Medicine, London, August 1985, pp. 958 ff", the conjugated gradient method
of Murdoch et al. in "J. Magn. Reson." 74, 226 (1987) and the
linearization of the Bloch equations according to Ngo et al. in "Magn.
Reson. Med." 5, 217 (1987).
These processes lead, to be sure, to excellent results. Those of Ngo et al.
approach rather close to an ideal signal response. The resulting
envelopes, however, require between 100 and 500 independent parameters for
their definition, so that their practical application becomes extremely
complicated.
An object of the present invention is, therefore, to present a process for
generating a spectrum of nuclear magnetic resonance signals by radiating a
sequence of n RF pulses onto a sample which is located in a homogeneous
static magnetic field, in which the envelope of the nuclear magnetic
resonance signal response in the frequency space is approximately a
rectangular function, in which, by a sequence of n RF pulses with as few
adjustable parameters as possible, a nuclear magnetic resonance spectrum
is generated with rectangular characteristics in frequency space.
This problem is solved according to the invention by the means that the n
RF pulses of the sequence are amplitude-modulated and their respective
amplitude distributions follow approximately the form of bell curves, and
that, in advance, the amplitude distribution of the entire pulse sequence
has been determined in a suitable optimization procedure under the
criterion of a minimizing of the deviation of the envelope of the nuclear
magnetic resonance signal response in the frequency space from a
rectangular function by variation of 3.multidot.n parameters
w.sub.k.sup.max, t.sub.k.sup.max, a.sub.k, in which w.sub.k.sup.max
signifies the relative amplitude of the k-th pulse of the sequence at the
position t.sub.k.sup.max of its extremum and a.sub.k, the width of the
k-th pulse.
The pulse sequence comprises a superposition of well known, easily
reproducible RF pulses, whose number n is limited to at most ten, which in
every case suffices for an optimal approximation of the envelope of the
signal response to a rectangular function. The total number of free
parameters to be controlled does not exceed 3.multidot.n.
The optimization of the sequence of bell curved shaped RF pulses according
to the invention occurs in advance in a manner known per se by variation
of the parameters w.sub.k.sup.max, t.sub.k.sup.max and a.sub.k
characterizing the bell shaped curves, which are varied until the
deviation of the envelope of the nuclear magnetic resonance signal
response in the frequency space from a rectangular function is minimal.
Since the optimization has to be carried out only once and then the
optimal parameters for the amplitude distribution w.sub.l (t) are
established, the rapidity of the optimizing procedure plays no role. In
principle, the parameters could also be determined by more or less
qualified guessing, by trial or by a simple iteration carried out "by
hand".
The minimization of the deviation of the envelope of the nuclear magnetic
resonance signal response in the frequency space from a rectangular
function is especially simply and conveniently accomplished by use of a
numerical fit program on an automatic data processing installation.
In an advantageous development of the method of the invention the
minimizing of the deviation of the envelope of the nuclear magnetic
resonance response in the frequency space from a rectangular function is
accomplished by optimizing of a given error function which describes the
deviation of the envelope from a rectangular function. The optimization
can be broken off when the error function exceeds a predetermined limit
value.
The bell curve shaped RF pulses of the sequence can generally also be
slightly asymmetrical. In a special form of performing the method of the
invention the sequence contains RF pulses whose amplitude distribution
follows the form of a Lorentz curve. Especially preferred is a form of
execution in which the amplitude distributions of the RF pulses follow the
form of Gaussian curves. The special parameter values mentioned below were
determined especially for Gaussian pulse cascades and led to especially
good results.
In a special development of the method of the invention the number of RF
pulses per sequence is n=3. Therewith there is yielded a pulse sequence
which is particularly well suited for the selective inversion of nuclear
magnetic moments.
The envelope of the inversion signal response, with use of the method of
the invention, comes particularly close to a rectangular function when the
parameters w.sub.k.sup.max, t.sub.k.sup.max /t.sub.p and
.DELTA.t.sub.k.sup.1/2 /t.sub.p take on in each case a corresponding value
from the interval
1,2<W.sub.2.sup.max /W.sub.1.sup.max <1,5; 0,3<W.sub.3.sup.max
/W.sub.1.sup.max <0,7
15<.DELTA.t.sub.1.sup.1/2 /t.sub.p <25; 15<t.sub.2.sup.1/2 /t.sub.p <25;
20<.DELTA.t.sub.3.sup.1/2 /t.sub.p <30; 20<t.sub.1.sup.max /t.sub.p <40;
40<t.sub.2.sup.max /t.sub.p 60; 70<t.sub.3.sup.max /t.sub.p <90
in which .DELTA.t.sub.k.sup.1/2 is half the pulse width of the k-th pulse
at the half pulse amplitude w.sub.k.sup.max /2 and a.sub.k
=ln2/(.DELTA.t.sub.k.sup.1/2).sup.2, especially when the parameters take
on in each case the value
W.sub.1.sup.max =-1,00; W.sub.2.sup.max =1,37; W.sub.3.sup.max =0,49;
.DELTA.t.sub.1.sup.1/2 /t.sub.p =18,9; .DELTA.t.sub.2.sup.1/2 /t.sub.p
=18,3; .DELTA.t.sub.3.sup.1/2 /t.sub.p =24,3;
t.sub.1.sup.max /t.sub.p =28,7; t.sub.2.sup.max /t.sub.p =50,8;
t.sub.3.sup.max /t.sub.p =79,5
With use of n=4 RF pulses in the pulse sequence w.sub.1 (t) the method of
the invention is especially well suited for the in-phase excitation of
transverse magnetization. An especially good result is achieved when the
parameters w.sub.k.sup.max, t.sub.k.sup.max /t.sub.p and
.DELTA.t.sub.k.sup.1/2 /t.sub.p take on in each case a corresponding value
from the interval
0,95<W.sub.2.sup.max /W.sub.1.sup.max <1,35; -1,7<W.sub.3.sup.max
/W.sub.1.sup.max <-1,3;
-0,7<W.sub.4.sup.max /W.sub.1.sup.max <-0,4;
15<.DELTA.t.sub.1.sup.1/2 /t.sub.p <20; 10<.DELTA.t.sub.2.sup.1/2 /t.sub.p
<15; 10<.DELTA.t.sub.3.sup.1/2 /t.sub.p <15;
10<.DELTA.t.sub.4.sup.1/2 /t.sub.p <20;
5<t.sub.1.sup.max /t.sub.p <25; 40<t.sub.2.sup.max /t.sub.p <60;
55<t.sub.3.sup.max /t.sub.p <75;
75<t.sub.4.sup.max /t.sub.p <95
in particular the values
W.sub.1.sup.max =0,62; W.sub.2.sup.max =0,72; W.sub.3.sup.max =-0,91;
W.sub.4.sup.max =0,33;
.DELTA.t.sub.1.sup.1/2 /t.sub.p =17,2; .DELTA.t.sub.2.sup.1/2 /t.sub.p
=12,9;
.DELTA.t.sub.3.sup.1/2 /t.sub.p =11,9; .DELTA.t.sub.4.sup.1/2 /t.sub.p
=13,9;
t.sub.1.sup.max /t.sub.p =17,7; t.sub.2.sup.max /t.sub.p =49,2;
t.sub.3.sup.max /t.sub.p =65,3;
t.sub.4.sup.max /t.sub.p =89,2
Especially advantageous is the use of the method of the invention as a
component of imaging methods, especially in NMR tomography, in
multidimensional NMR spectroscopy and especially for volume-selective NMR
spectroscopy. The sequences of n RF pulses according to the invention can
be part of a NOESY pulse sequence or part of a COSY pulse sequence.
The invention relates also to an NMR spectrometer with an RF pluse
generator and an arrangement for the drive of the RF pluse generator, the
arrangement comprising a memory for the storage of data sets for the
generation of RF pluse sequences wherein the memory comprises a data set
for the generation of a pulse sequence with a number n of at least 2 but
not more than 10 amplitude modulated bell shaped RF pulses, each RF pulse
comprising an amplitude w.sub.k.sup.max, an extremum position
t.sub.k.sup.max, and a width a.sub.k and, in advance, the amplitude
distribution of the entire pulse sequence has been determined in a
suitable optimization procedure under a criterion of a minimization of a
deviation of an envelope of a nuclear magnetic resonance signal response
in frequency space from a rectangular function by variation, for each
pulse, of w.sub.k.sup.max, t.sub.k.sup.max, and a.sub.k. Such a
spectrometer can be further developed so that the above-described
developments of the method of the invention can be carried out on it.
The invention is described in detail with the aid of the embodiments
represented in the drawings. The features to be learned from the
specification, the tables and drawings can be used in other embodiments of
the invention individually by themselves or collectively in arbitrary
combination.
FIG. 1 shows Fourier transforms (in broken lines) together with the
numerically calculated transverse magnetization <M.sub.xy > (solid line)
of Gaussian pulse cascades, whose three individual pulses yield the
nominal flip angle +45.degree., -90.degree., +135.degree., in which
(a) each individual pulse has a width proportional to the flip angle; or
(b) the individual pulses have equal widths and amplitudes proportional to
the flip angle.
FIG. 2 shows a comparison of the M.sub.z responses of the inversion pulses
whose envelope (is)
(a) a single Gaussian pulse;
(b) a Sinc pulse; and
(c) a Gaussian pulse cascade according to the invention with n=3.
FIG. 3 shows the M.sub.z responses to an inversion pulse cascade according
to the invention proceeding from a start function
(a) as well as after various optimization cycles (b), (c), (d).
FIG. 4 shows:
(a) the start function with n=4 and the corresponding M.sub.x - and M.sub.y
- profiles after an in-phase excitation; and
(b) the optimized excitation cascade and the corresponding optimized
signals.
FIG. 5 shows:
(a) the experimental result of an inversion cascade n=3 (above), as well as
the result of a simulation of this experiment (below);
(b) the experimental result of an excitation cascade n=4 (above), as well
as the corresponding simulation calculation (below); and
FIG. 6 shows a comparison of the simulated behavior
(a) of a 90.degree. Gaussian pulse;
(b) of a 270.degree. Gaussian pulse; and
(c) of an excitation pulse cascade according to the invention (n=4).
FIG. 7 shows a schematic diagram of an NMR spectrometer according to the
invention.
For small flip angles the frequency response of the spin system corresponds
approximately to the Fourier transform of the envelope of the pulse in the
time domain. Far off-resonance where the magnetization trajectories are
effectively confined to small excursions in the vicinity of the "north
pole" of the reference system, it has been shown that the above
equivalence holds even for large effective flip angles. For the prediction
of the on-resonance behavior in the case of large flip angles such as are
required for inversion, refocussing or even for efficient excitation of
the transverse magnetization, the equivalence, however, completely
collapses. Although, therefore, pulses with Gaussian or Sinc envelopes may
have useful properties, these should not be attributed to the fact that
the Fourier transforms of these envelopes have the form of Gaussian and
rectangular functions in the frequency domain.
In the following there is introduced a new class of amplitude modulated
pulses which are useful especially both for the selective inversion, as
well as for in-phase excitation, and which are described by the analytical
form:
##EQU1##
This function represents an envelope of the time duration t.sub.p and
consists of a superposition of n Gaussian functions, in which for the k-th
Gaussian function the position of the extremum is given by
t.sub.k.sup.max, the peak amplitude by w.sub.k.sup.max and the width by
a.sub.k =ln2/(.DELTA.t.sub.k.sup.1/2).sup.2 [ 1a]
and .DELTA.t.sub.k.sup.1/2 is half the pulse width of the k-th pulse in
the half pulse amplitude. The function .pi. (0, t.sub.p) represents a box
function with the value 1 between t=0 and t.sub.p and the value 0
everywhere otherwise.
Since the Fourier equivalence is valid for large frequency offsets from
resonance, the Fourier transforms of an ideal pulse cascade must lie
outside the relevant band width as near as possible to 0. Abrupt
variations of the amplitude, therefore, should be avoided, because they
lead to erratic Fourier transforms. For this reason the pulse forms of the
individual pulses in the cascade were chosen as Gaussian functions in
contrast to the otherwise usual rectangular envelopes. Furthermore, the
form of the Fourier transform of a pulse is influenced not by the
amplitude but only by its width. As direct consequence of the addition and
displacement theorems it can be shown that a cascade of n shifted Gaussian
pulses with identical widths (a.sub.k =a for all k) and variable
amplitudes has a Fourier transform that consists of a sum of n Gaussian
envelopes with identical widths (and variable amplitudes):
##EQU2##
The frequency-domain Gaussian pulses will therefore simply be associated
with different phases which reflect the time shifts in the cascade. From
this it can be perceived that the output sequence for a Gaussian cascade
should have individual pulses with equal widths a.sub.k and amplitudes
.OMEGA..sub.k.sup.max which vary according to the desired nominal flip
angles. In this manner, it can be expected that the excitation will
decline like that of a single Gaussian pulse.
For the demonstration of this argument, FIG. 1 shows the Fourier transforms
together with numerical simulations of the <M.sub.xy > responses to two
Gaussian pulse cascades whose individual pulses in each case have nominal
on-resonance flip angles of +45.degree., -90.degree. and +135.degree.. If
the amplitudes are constant and the widths are proportional to the flip
angles, it is obvious that the tails in the excitation profiles go very
slowly to zero. In contrast to this, a cascade of Gaussian pulses with
equal pulse widths, but amplitudes proportional to the flip angles leads
to a response which decays very much more rapidly. One should also note
that the Fourier transform does not directly predict the on-resonance
behavior. Although the principle of amplitude modulation in contrast to
modulation of the pulse duration can be extended easily to pulse cascades
with arbitrary phase shift, at the moment there should be considered only
pulse trains with constant phase, in which, to be sure, phase reversal
(for example, from x to -x) is to be allowed. The pulse responses in FIG.
1 are represented as a function of the reduced frequency offset
.OMEGA./w.sub.1.sup.max, in which .OMEGA.=W.sub.0 -W.sub.RF and
w.sub.1.sup.max are the frequency offset and the maximal RF amplitude of
the 90.degree. pulse.
The pulses were optimized on the computer with a modified Simplex
procedure, in which the deviation from a rectangular target function was
minimized. It should be noted that the above mentioned Fourier conditions
are sufficient for the desired off-resonance behavior, but are by no means
necessary. It can well be that cascades with unequal pulse widths a.sub.k
likewise lead to good quality response signals. For this reason the
optimization was carried out with all the independent parameters of
equation (1). The minimizing of the error of a pulse response which
satisfies the Bloch equations is rendered especially difficult by the
presence of a large number of shallow false minima. This problem is solved
most simply by redefinition of the form of the error function during the
optimization procedure.
As a starting value for a pulse cascade with three pulses, the following
sequence was chosen:
G{270.degree..sub.-x } G{270.degree..sub.x } G{180.degree..sub.x }[3]
The sequence consists of a simple preparation cycle, which is followed by
an ordinary inversion pulse. The G(270.degree..sub.-x)
G(270.degree..sub.x) cycle has no net effect in the region of the
resonance, but prepares the off-resonance magnetization in a favorable
manner for the subsequent action of the 180.degree. pulse for the
generation of a rectangular inversion profile. In view of the
considerations in regard to the Fourier transforms, all three Gaussian
pulses were chosen with the same width but with amplitudes proportional to
the flip angles.
In FIG. 2, the M.sub.z magnetization as a function of the standardized
frequency shift is compared after application (a) of a 180.degree.
Gaussian pulse, (b) a 180.degree. Sinc pulse and (c) the cascade according
to equation (3). The Gaussian pulse was cut off at 5% of w.sub.1.sup.max,
the Sinc pulse had four zero crossings on each side of the maximum and the
cascade had the parameters of cycle 0 in Table 1. Although the result of
the cascade of FIG. 2C within the chosen band width is anything but
perfect, the missing excitation outside of the desired band width shows an
improvement over the individual Gaussian pulse and the Sinc function. The
sequence G(270.degree..sub.-x) G(270.degree..sub.x) G(180.degree..sub.x)
was used, therefore, as a starting point for an optimization.
FIGS. 3A to D show the M.sub.z magnetization profiles after successive
cycles in the optimization procedure. There, (a) is the start function,
(b), (c) and (d) are the results of the first, second and third
optimization cycle, the latter being the finally accepted inversion
cascade G.sup.3. The parameters for the resulting pulses are presented in
Table 1. For each optimization cycle (150 iterations) there are defined
two error functions f.sup.in and f.sup.out, one for the isochromates
within and one for the isochromates outside the excitation and inversion
rectangles, respectively:
##EQU3##
There, M.sub.k is the magnetization response for t=t.sub.p at the k-th
offset and 1 is the point at which the M.sub.z response runs through 0
(only for inversion). The intervals m and m' corresponding to a small
transition zone, in which the response can move freely without
contributing to the error function. The error functions are defined as
f.sub.inv.sup.out (M.sub.k)=f.sub.exc.sup.out (M.sub.k)=(M.sub.kx.sup.2
+M.sub.ky.sup.2).sup.a [ 6]
f.sub.inv.sup.in (M.sub.k)=b(1+M.sub.kz) [7]
f.sub.exc.sup.in (M.sub.k)=c[(1-M.sub.kx).sup.2 +dM.sub.ky.sup.2 ].sup.g
+h(M.sub.kz.sup.2 +M.sub.ky.sup.2) [8]
in which the subscripts inv and exc relate to functions that are used for
the optimization of inversion and excitation pulses, respectively. The
parameters a-h, which define the error functions according to equations
(6) to (8) can be changed from cycle to cycle during the optimization and
are given in Tables 1 and 2 together with the values m and m' as well as
the parameters for each individual pulse in the cascade, the latter
parameters being represented in the form of the position t.sub.k.sup.max
of the k-th pulse in units of the time duration t.sub.p of the sequence,
the relative amplitude w.sub.k.sup.max and the line width
.DELTA.t.sub.k.sup.1/2 likewise in units of t.sub.p. Let it be heeded
that in each case either c or h in equation (8) is zero. The optimization
was carried out for 80 isochromates and each response was calculated by
obtaining a numerical solution of the Bloch equations with 200 equal time
intervals in the integration. The frequency offsets are chosen so that 1
lies at about the 25th isochromate, and the error function is normalized
in order to take into account any variations in 1 from iteration to
iteration. Each optimization cycle consists of 150 iterations of a Simplex
minimization of the error function.
In FIG. 3A to D there are also shown the pulse forms that are yielded after
each cycle, so that it is possible to make visible the gradual
optimization of the pulse. The last inversion pulse, which is called
G.sup.3, has the parameters given in Table 1 and is rather similar to the
initial value in cycle 0. To obtain the present result only, three cycles
of the optimization procedure are needed, which require approximately 15
minutes of c.p.u. time on a VAX 8550 cluster rated at 14 mips.
The G.sup.3 cascade can be used not only for the inversion, but also as a
refocusing pulse. Although the pulse was not optimized for this purpose,
it nevertheless performs well also in this respect. This property is used
in the following for the build-up of excitation pulses.
A further aim is to find a pulse which in a selected frequency interval can
convert the longitudinal magnetization into transverse magnetization with
the least possible phase dispersion. Outside the chosen interval the
output state should be disturbed as little as possible. These requirements
can be summarized in the expression "rectangular in-phase excitation".
Hitherto, no suitable pulse cascade with only three Gaussian pulses has
yet been found for this purpose. The simplest four-pulse cascade is a
usual type 270.degree. Gaussian excitation pulse, upon which there follows
a G.sup.3 -cascade as a refocusing pulse.
G{270.degree..sub.x }G.sup.3.sub.x [ 9]
This cascade was used as starting value. The phase of a cascade is to be
indicated in the following by a subscript which represents the phase of
the first individual pulse. The total on-resonance flip angle of the
sequence in equation (9), therefore, is +90.degree.. The course of the
optimization is shown in Table 2. Since excitation by its nature presents
a more difficult problem, the optimization converged more slowly and
required 11 cycles and about 90 minutes of c.p.u. time.
FIG. 4 shows the responses to the start pulse (a) and to the final pulse
(b), as it was generated by the optimization. The latter is called G.sup.4
in the following. If one defines, as a useful band width the range in
which the amplitude M.sub.xy exceeds 90% of the maximal value, the maximal
phase deviation of the G.sup.4 pulse in this window amounts to less than
5.degree..
Experiments with Gaussian pulse cascades were carried out on a Bruker
AM-400 spectrometer which was outfitted with an arrangement for selective
excitation of Oxford Research Systems (Series No. H 1657/0013). The pulse
shapes were generated by a Pascal program which runs on the Aspect 3000
computer, which generates a data file that can be used by the Bruker
SHAPE-PACKAGE, in order to create data for the wave-form memory. The
generation of the enveloping functions in the case of inversion or
excitation is reduced there to the simple entering of the nine or twelve
relevant parameters, which are given in each case at the end of Table 1 or
2, respectively.
The little boxes inserted in FIGS. 1 to 3 as well as the left boxes in FIG.
4 show the envelope of the Gaussian pulse cascade used in each case.
FIG. 5A shows the experimental behavior of the G.sup.3 cascade. The results
are obtained by applying the sequence
G.sup.3 .PHI.-90.degree.-acquisition
in which the 90.degree. pulse is an ordinary hard nonselective pulse. The
position of the proton resonance of benzene (doped with Cr(acac).sub.3)
was changed by stepwise incrementing of the transmitter frequency. The
phase of the G.sup.3 cascade was turned through .PHI.=0.degree.,
90.degree., 180.degree. and 270.degree., while the receiver phase was held
constant in order to assure that only the Z-component of the magnetization
generated by the G.sup.3 cascade was measured. The agreement with the
theory is excellent. In the upper diagram of FIG. 5 there is shown the
actual experimental result, while the lower diagram of FIG. 5 shows the
simulation of this experiment. The incomplete inversion in the center is a
manifestation of the RF inhomogeneity in the sample, which is represented
by a Lorentz distribution of the flip angles with a half-value width of
w.sub.1 =0.12.multidot.w.sub.1.sup.nominal. The RF inhomogeneity has only
a slight effect on the resulting profiles, which was likewise predicted
from the theory.
FIG. 5B shows a similar set of results for the G.sup.4 excitation cascade.
These were obtained by the use of a simple G.sup.4 acquisition sequence
and stepwise incrementing of the frequency of the transmitter, as
described above. Again, the agreement of theory and experiment is nearly
exact. No frequency-dependent phase correction was performed on these
spectra. One should heed that the pulse response has a constant phase up
to the limit of the excitation band width.
For comparison, FIG. 6 shows the simulated behavior (a) of a single
90.degree. Gaussian pulse, (b) of a 270.degree. Gaussian pulse and (c) of
the G.sup.4 excitation cascade. It is evident that the cascade presents a
considerable improvement both for the profile itself and also for the
phase properties of the pulse response.
The method of the invention, especially the use of the G.sup.3 inversion
cascade and the G.sup.3 excitation cascade, can also be used as part of
imaging methods in NMR tomography. Since the pulse sequences of the
invention can be generated easily experimentally, also their use in
multi-dimensional NMR spectroscopy, especially in volume selective NMR
spectroscopy, is advantageous. The sequence of n RF pulses according to
the invention can, in particular, also be part of a pulse sequence for
correlated NMR spectroscopy (COSY) or of a NOESY pulse sequence.
Generally, instead of the Gaussian pulses described in detail above other
RF pulses, with substantially bell-curve shaped amplitude distribution
which may even be slightly asymmetrical, can also be used as components of
the RF pulse sequence according to the invention. Such a bell curve form
is described, for example, also by the well known Lorentz function:
S.sub.k (t)=w.sub.k.sup.max /.pi..multidot.a.sub.k /[a.sub.k.sup.2
+(t-t.sub.k.sup.max).sup.2 ]
FIG. 7 is a schematic diagram of an NMR spectrometer 1 in accordance with
the invention, comprising an RF pulse driver 2 containing a memory 3, and
driving an RF pulse generator 4.
TABLE 1
______________________________________
Error function Pulse No. (k)
Cycle m m' a b 1 2 3
______________________________________
0 t.sub.k.sup.max /t.sub.p
16.6 50.0 83.3
w.sub.k.sup.max
-1.00 1.00 0.66
.DELTA.t.sub.k.sup.1/2 /t.sub.p
16.0 16.0 16.0
1 2 2 1/2 15 t.sub.k.sup.max /t.sub.p
23.3 45.8 73.8
w.sub.k.sup.max
-1.00 1.10 0.68
.DELTA.t.sub.k.sup.1/2 /t.sub.p
16.8 20.4 15.7
2 2 2 1/2 5 t.sub.k.sup.max /t.sub.p
27.5 48.6 77.3
w.sub.k.sup.max
-1.00 1.04 0.59
.DELTA.t.sub.k.sup.1/2 /t.sub.p
18.6 20.9 20.0
3 2 2 1/2 15 t.sub.k.sup.max /t.sub.p
28.7 50.8 79.5
w.sub.k.sup.max
-1.00 1.37 0.49
.DELTA.t.sub.k.sup. 1/2 /t.sub.p
18.9 18.3 24.3
______________________________________
TABLE 2
__________________________________________________________________________
Error function Pulse No. (k)
Cycle
m m'
a c d g h 1 2 3 4
__________________________________________________________________________
0 t.sub.k.sup.max /t.sub.p
14.2
49.1
64.9
85.3
w.sub.k.sup.max
1.00
0.64
-0.87
-0.31
.DELTA.t.sub.k.sup.1/2 /t.sub.p
12.2
13.5
13.1
17.42
1 2 2 1 0 0 0 3 t.sub.k.sup.max /t.sub.p
13.4
49.9
64.7
85.2
w.sub.k.sup.max
0.97
0.65
-0.86
-0.35
.DELTA.t.sub.k.sup.1/2 /t.sub.p
12.1
14.1
12.6
15.7
2 2 2 1 0 0 0 6 t.sub.k.sup.max /t.sub.p
13.2
49.8
64.3
85.0
w.sub.k.sup.max
0.93
0.67
-0.92
-0.33
.DELTA.t.sub.k.sup.1/2 /t.sub.p
12.7
14.1
12.3
15.7
3 4 2 1 1 6 1 0 t.sub.k.sup.max /t.sub.p
13.5
50.2
65.0
85.0
w.sub.k.sup.max
0.97
0.75
-0.97
-0.36
.DELTA.t.sub.k.sup.1/2 /t.sub.p
12.9
13.2
12.2
15.6
4 3 2 1 1 6 1 0 t.sub.k.sup.max /t.sub.p
18.1
50.2
65.6
87.5
w.sub.k.sup.max
0.81
0.71
-1.02
-0.30
.DELTA.t.sub.k.sup.1/2 /t.sub.p
13.6
13.1
10.9
14.1
5 2 2 1 1 6 1 0 t.sub.k.sup.max /t.sub.p
18.7
50.7
66.1
87.7
w.sub.k.sup.max
0.76
0.75
-0.99
-0.36
.DELTA.t.sub.k.sup.1/2 /t.sub.p
14.6
14.1
12.0
13.7
6 1 1 1 1 6 1 0 t.sub.k.sup.max /t.sub.p
16.5
49.4
65.3
89.3
w.sub.k.sup. max
0.70
0.72
-0.95
-0.37
.DELTA.t.sub.k.sup.1/2 /t.sub.p
16.3
15.1
13.2
14.5
7 1 1 1/2
1 6 1/2
0 t.sub.k.sup.max /t.sub.p
16.7
49.2
65.2
89.8
w.sub.k.sup.max
0.63
0.74
-0.91
-0.35
.DELTA.t.sub.k.sup.1/2 /t.sub.p
17.4
14.0
13.1
14.7
8 1 1 1/2
1 3 1/2
0 t.sub.k.sup.max /t.sub.p
17.6
49.3
65.4
89.7
w.sub.k.sup.max
0.63
0.75
-0.89
-0.35
.DELTA.t.sub.k.sup.1/2 /t.sub.p
17.7
13.7
13.1
14.7
9 1 1 1/2
1 0.5
1/2
0 t.sub.k.sup.max /t.sub.p
16.6
48.6
64.7
88.9
w.sub.k.sup.max
0.58
0.74
-0.85
-0.34
.DELTA.t.sub.k.sup.1/2 /t.sub.p
17.7
13.6
12.7
14.4
10 2 2 1 4 0.71
1/2
0 t.sub.k.sup.max /t.sub.p
17.2
48.9
65.05
89.1
w.sub.k.sup.max
0.59
0.72
-0.88
-0.34
.DELTA.t.sub.k.sup.1/2 /t.sub.p
17.6
12.9
12.1
13.4
11 2 2 1 4 0.84
1/2
0 t.sub.k.sup.max /.sub.p
17.7
49.2
65.3
89.2
w.sub.k.sup.max
0.62
0.72
-0.91
-0.33
.DELTA.t.sub.k.sup.1/2 /t.sub.p
17.2
12.9
11.9
13.9
__________________________________________________________________________
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